Abstract:We consider a two-sided, finite-horizon model of search and matching with heterogeneous types on both sides of the market. The quality of the pool of potential matches deteriorates as agents who have found mutually agreeable matches exit the market. With automatic participation of all agents in each round, the market performs a sorting function in that more attractive types of agents have a chance to meet and match with their peers in earlier rounds of the market. If agents incur an arbitrarily small cost in order to participate in each round, however, the market completely loses its sorting function as all agents rush to participate in the first round and match with anyone they meet.